A cluster state can be loosely defined as an entangled set of qubits arranged in a lattice. Breigel and Raussendorf strictly define a cluster state as “Let each lattice site be specified by a d-tuple of (positive or negative) integers a ε Zd. Each site has 2d neighboring sites. If occupied they interact with the qubit a”. This implies that a cluster state has interactions between all nearest neighbor qubits. In one dimensions (d=1) this results in a linear chain of qubits, of arbitrary length with each qubit connected to both of its nearest neighbors. All of the internal qubits will have two interactions while the end qubits will have one. Such a one dimensional nearest neighbor cluster state has been shown to be capable of several interesting applications, presuming the cluster state is “long” enough. Of more interest are two dimensional cluster states which have been shown to be a universal resource for quantum computation, if the cluster state is “large” enough. Even for small systems, two dimensional cluster states are more desirable as they are able to implement more diverse and complex calculations.
Traditional generation of a cluster state consists of an optical table several meters on each side. On this table is a high power pump laser system such as a pulsed Ti:Sapphire laser. The pump beam is incident on a nonlinear material such as BBO, BiBO or PPKTP etc. The photons from the pump then have a small chance to undergo Spontaneous Nonlinear Parametric Down Conversion (SPDC) to create an entangle pair of photons, called signal and idler photons. Alternative means of photon generation are equally valid such as four wave mixing (FWM).
To create larger clusters the pump passes through multiple nonlinear materials (a cascade configuration) or is reflected back onto the material (a multi-pass configuration). These methods can create multiple simultaneous independent pairs of qubits. To create one large cluster state the pairs are sent through (i.e. acted on by) an entangling operation. Normally the controlled phase gate (CPhase) or equivalently controlled Z gate (CZ) is used in the state of the art. The simplest and most efficient means of implementing the general CZ gate requires 3 bulk optical asymmetric beam splitters in a specific alignment. These operations are effectively performed in parallel with each qubit entering and exiting in its own port and the order of the operations is irrelevant if CZ is used. Once all the entangling operations are successfully completed the cluster state is fully constructed and any Measurement Based Quantum Computing (MBQC) algorithm can be implemented as a sequence of single qubit rotations and measurements on each qubit in a predetermined sequence. In the state of the art, cluster states are created from simultaneously generated qubits in parallel modes rather than from sequential qubits in a single mode. This is in large part due to the spontaneous nature of single photon sources. It is impossible to predict the time between two subsequent spontaneous events.